In contrast to the elliptic complex obtained for the case of the topologically twisted theory [1]}, the complex (REF ) turns out to only be transversally elliptic, which we show in due course. Note that, at the fixed points of \(v\) , the complex indeed splits into the ASD/SD complex and the scalar Laplacian as
\(\Bigl (\begin{tikzcd}[column sep=small]0[r] & \Omega ^1_X(\mathfrak {g}_P)[r,"\mathrm {d}^\pm \oplus \mathrm {d}^\dagger "] & \Omega _X^{2\pm }(\mathfrak {g}_P)\oplus \Omega ^0_X(\mathfrak {g}_P)[r] & 0\end{tikzcd}\Bigr )\oplus \Bigl (\begin{tikzcd}[column sep=small]0[r] & \Omega ^0_X(\mathfrak {g}_P)[r,"\Delta "] & \Omega ^0_X(\mathfrak {g}_P)[r] & 0\end{tikzcd}\Bigr ).\) 
